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## Hadrons and Quarks, an Introduction to the Theory of Mesons

and Baryons

Walter Pfeifer

Switzerland

38 pages, 10 line figures

2010

### Preface

Since fifty years physicists are convinced that protons, neutrons and related
particles (hadrons) consist of subparticles named quarks. The properties of the
quarks determine the feature of the hadrons. Although the quarks cannot be
detected as free particles their configuration in the hadrons is known in detail.

In this book we restrict ourselves to low-mass hadrons and low-mass quarks.
Higher states can be dealt with using the same technique. Here, a good agreement
between theoretical and measured masses and magnetic moments of hadrons is
presented.

The detailed developments and the numerous references to preceding places make
this book easy to follow. However, knowledge of elements of quantum mechanics
and Lie algebra is a prerequisite.

As in our previous publication, Pfeifer, W., 2008, we utilize the following
symbols: operators are written in bold letters and three-dimensional vectors are
marked with an arrow.

### Contents

Preface 2

1 Hadrons 4

1.1 Characteristics of elementary particles 4

1.2 Conservation laws 5

1.3 Compiling of the properties of low-mass hadrons 6

1.4 Diagrammatic arrangement of hadrons 10

2 Quarks 13

2.1 Low-mass quarks 13

2.2 Mesons and quarks 14

2.3 Baryons and quarks 16

2.4 The isospin of the quark configurations, mesons 18

2.5 Meson representation with spins 20

2.6 Quark representation of baryons allowing for isospin and spin 21

3 Masses and magnetic Moments of Hadrons and Quarks 26

3.1 Masses of mesons and effective masses of quarks 26

3.2 Masses of baryons and effective masses of quarks 28

3.3 Magnetic moments of baryons and quarks 31

4 Epilogue 36

References 37

Index 38

### 1 Hadrons

Up to the year 1932 the proton and the
neutron were the only known heavy elementary particles
. They are named nucleons
. Scattering experiments show that they attract themselves
extremely strongly when they are very near (nearly touching themselves)
independently oft their charges. One says that they interact strongly.

In the following years, investigating
cosmic rays and using accelerators, other particles with strong interaction were
found. Particles of this type being lighter than nucleons were named mesons
. Most of the heavy ones are called baryons. Mesons and
baryons
constitute
the collective of the hadrons.

In the next section we deal with
characteristic quantum numbers and data of hadrons.

### 1.1 Characteristics of elementary particles

The quotient of the **charge**
and the **mass**
of a free moving particle,
, is measured using a combination of magnetic and electric
fields. We will give the charge as a multiple
of the elementary charge
including the sign. Because for free particles
is an integer, experimentalists can determine it
and calculate the mass
.

Most elementary particles (including
hadrons) are endowed with as **spin**. As in atomic or nuclear physics the
eigenvalue of the square of the spin operator
is
, where
is a half-integer or integer (including zero)
value. The
-component
meets
. For
and
we will use the symbol
and for
we take
.

The **isospin**
is a virtual spin. To the nucleons W.
Heisenberg assigned the isospin
. The proton is characterized by the component
and the neutron by
. Both nucleons which have nearly the same mass, form an
isospin doublet
. Three hadrons with about the same mass are summarised to a
triplet
with isospin
. The member with positive charge has
, the neutral member has
and negative charge means
. If a hadron with charge zero has no partner with comparable
mass, it is interpreted as a charge singlet with
.

Isospins of interacting particles are
couplet with the same formalism as spins using Clebsch-Gordan
coefficients.

For hadrons, Gell-Mann and Nishijima have
introduced the **strangeness**
**
**as a quantum
number as follows

The quantum number
(baryon charge
) is 1 for baryons and 0 for mesons. The fact that several
hadrons are generated by strong interaction and decay by weak interaction first
was denoted as a strange property. For these particles
holds.

Starting from the strangeness, the **hypercharge**
is defined as follows

It’s true that the marking hypercharge isn’t accepted
generally, but the quantity
is useful for the geometrical representation of
hadrons (section 1.4).

Originally, **parity**
gives the sign change of the wave function in
question under reflection with regard to the origin of the spatial coordinates.
The parity of elementary particles mostly is determined starting from known
parities of interacting particles. The conservation law for parity (section 1.2)
is applied. Per definition, nucleons have positive parity and
-mesons (in ground state) has negative parity.

An **antiparticle**
has the same mass, spin and isospin as the
corresponding particle. The sign of its charge, of its baryon charge
and of its isospin component
is inverted. Therefore the hypercharge
has a different sign.

### 1.2 Conservation laws

The total (relativistic) **energy**
of a moving particle with rest mass
and (classic) impulse
reads

.

For
we have
. In a reaction or in a decay the sum of the total energies of
all involved particles is constant.

The total **charge**
i.e. the sum of the charges (taking into
account their signs) of the involved particles is conserved.

The **spin****
** of a system of
particles is the vectorial sum of the individual spins formed using the addition
rules of quantum mechanics (cf. Pfeifer, 2008, p. 43). The magnitude and the
z-component of the spin are conserved.

The **isospin**
of a system is determined with the same
formalism as the spin. If the particles have strong interaction the isospin and
its component
are conserved (which isn’t true for
electromagnetic and weak interactions)

For strongly interacting particles (e.g. hadrons) the sum of
the **baryon charges****
**is also a
conserved quantity.

Because the algebraic expression for the **strangeness**
(equation 1.1.1) is a linear combination of
conserved quantities, the strangeness is also conserved.

The **parity**
of a system of particles is the product of the
parities (+1 or -1) of the constituents. This parity is conserved in reactions
or decays.

### 1.3 Compiling of the properties of low-mass hadrons

Table shows the measured properties of
the low-mass mesons with spin zero and negative parity. They are named
pseudoscalar mesons. Most of the lifetimes of the given particles are near
s. For three mesons which decay in two
they are much shorter.

Table 1.3.1. Properties of the pseudoscalar
mesons

From Greiner, W., Müller, B., 1994, p.233

The table contains the
particle-antiparticle pairs
. The hypercharge
is calculated with equation 1.1.2. Basing on
equation 1.2.1 the mass is given in the unit
.

The next table contains mesons with spin 1
and negative parity. They are named vector mesons.

Table 1.3.2. Properties of vector mesons

From Greiner, W., Müller, B., 1994, p. 242.

There are also other meson multiplets with
higher spins and masses.

The next table shows the
properties low-mass baryons
. As is known, the proton is stable, the neutron has a
lifetime of 15 min and most of the given baryons live near
s long. Only the baryon
which emits a
-quant has a much shorter lifetime.